Let us elaborate more on the notion of the noncommutative
value as a description of the full quantum information involved.
A naive thinking about the full information should, for example,
encode the full statistical distribution of the corresponding
projective measurements of the observable for a fixed state.
That has information about the expectation value as well as the
quantum fluctuations around it, including the Heisenberg uncertainty.
That may give a good idea on the quantum amount of translation
\([\hat{x}_{\!\scriptscriptstyle B}]_\phi \) , only which will be able to reveal the
very interesting quantum features of the transformations studied
in Refs.[1]}, [2]}. Though the latter articles do not explicitly discuss
such a quantum value translated, the description of the effects of
the translations as in how specific kinds of states are transformed
are presented, illustrating the `quantumness' of the change in the
translated position as involving changes in the quantum fluctuations
and even entanglement. Looking at the individual results on the
changes of the wavefunctions, the quantum value of change is
implicitly there. The notion of a noncommutative value of a
quantum observable we introduced recently [3]}, [4]}, is
exactly a concrete mathematical way to describe the kind of
quantum values. In fact, the quantum spatial translation may be
one of the best place to reveal the nature of the noncommutative
values as formulated from abstract mathematics. We apply a new
variant of the formulation of such noncommutative values, which
best suits the purpose at hand, below to look at the quantum
spatial translations of Refs.[1]}, [2]}, aiming at understanding
better both the quantum reference frame transformations and
the noncommutative values.
